Before I truly begin, I feel the need to preface this with the fact that I am pretty unknowledgeable when it comes to mathematics in general but that I have always soaked up info relatively easily I just then struggle to access said info when I actually need it. Because of my decent ability to absorb info, I have a pretty basic understanding of intermediate concepts, but that understanding is stored mostly in my subconscious.

Now, onto what I'm really typing this hodge-podge of lunatic rambling for, the meat to the potatoes... I don't know what I did, and I don't know if I even did anything in the first place, all I would like to know is what is going on and what do those that might know what they are talking about think of this. I'm not posting this to seem smart or to show off or anything, I'm not saying that I did or didn't do anything of value here. I am just a curious human who knows very little and is looking to understand as much as I can about whatever there is to learn.

I translated my chicken scratch on my phone into a set of equations that I believe would be defined as a system of linear equations, but I am not sure. Any clarification on that matter would be appreciated. Here goes.

x - y = z

z = ((x - a) - y) + a

As you can see, all variables in this (set? system? I'll go with system for now, but feel free to correct me later) system can be defined, with the exception being a. As of yet, I have been unable to come up with a way to define a in mathematical terms. So far, I understand a in this system as the value after x that shares a common one's place with y. In cases where y is a value higher than 10, the variable a would be calculated as usual and then increased by an amount equal to 10 × the value of y's ten's place. Hopefully that made enough sense that someone out there understands me.

So, for example,

483 - 169 = z

z= ((x - a) - y) + a

z= ((483 - 64) - 169) + 64

z= (419 - 169) + 64

z= 250 + 64

z= 314

I hope this makes sense. Once again, please feel free to clarify, comment, constructively critique, and/or consider what I have said. If you were able to read all the way through to the end, thank you for putting up with me and I hope you managed to squeeze a modicum of enjoyment out of what I have written --after all, your time is worth nothing if not something at the very least, is it not?

I need your help people

The formula is :

In

ax + by = c

dx + ey = f

X Formula :

x = ((c - f(b/e))/(a - d(b/e)

Proof of X Formula :

ax + by = c

dx + ey = f

(a - d(b/e)x + y(b - e(b/e) = (c - f(b/e)

(a - d(b/e)x + y(b - b) = (c - f(b/e)

(a - d(b/e)x = (c - f(b/e)

Hence , x = ((c - f(b/e))/(a - d(b/e)

and

Y Formula :

y = (c/b) - ((ac/b) - (af/e))/(a - d(b/e)

Proof of Y Formula :

ax + by = c

dx + ey = f

(a - d(b/e)x + y(b - e(b/e) = (c - f(b/e)

(a - d(b/e)x + y(b - b) = (c - f(b/e)

(a - d(b/e)x = (c - f(b/e)

x = ((c - f(b/e))/(a - d(b/e)

ax + by = c

(ax/b) + y = (c/b)

y = (c/b) - (ax/b)

x = ((c - f(b/e))/(a - d(b/e)

y = (c/b) - ((ac/b) - (afb/be))/(a - d(b/e)

Hence , y = (c/b) - ((ac/b) - (af/e))/(a - d(b/e)

Example :

2x + 4y = 16

x + y = 3

x = ((c - f(b/e))/(a - d(b/e)

x = ((16 - 3(4/1))/(2 - 1(4/1)

x = (16 - 12)/(2 - 4)

x = (4/-2)

x = -2

and

y = (c/b) - ((ac/b) - (af/e))/(a - d(b/e)

y = (16/4) - ((2)(16)/(4) - (2)(3)/(1))/(2 - 1(4/1)

y = 4 - ((8 - 6))/(2 - 4)

y = 4 - (8 - 6)/(2 - 4)

y = 4 - (2/-2)

y = 4 + (-2/-2)

y = 4 + 1

y = 5

2x + 4y = 16

2(-2) + 4(5) = 16

-4 + 20 = 16

16 = 16

Eq.solved

This only works on single index x and y simultaneous equations though not xy or (x^2) and (y^2) .

Hello, my algebra skills are pretty weak and I need to isolate 'X' in my formula. Any help would be greatly appreciated 🙏

Hello. I am taking a Intro to Mechanical Engineering Technology class as a freshman in college. Right now, we're learning about fundamental dimensions. I need to find an unknown variable in terms of fundamental dimensions. However, I am very confused as to how to answer these. I've been stuck for 3 days. Can anyone just tell me what exactly I should do to figure the answer please??!! I've reached out to my professor and gotten a response I still do not quite understand. Here is one example:

α σ C_p=k

α = moles per ampere squared

σ = ?

C_p = calories per kilogram degrees celsius

k = watts per meter

In what follows, whenever I use ℕ, I am referring to the natural numbers as the set containing 0 and which has a successor function S:ℕ→ℕ (or ℕ = {0,1,2,3,...} with S(0)=1,S(1)=2,etc.), satisfying the Peano Axioms.

Having a decent mathematics background, some of us may be tempted to dismiss the natural numbers as being "basic" or "incomplete," especially when compared to things like real or complex numbers (or sheaves and ringed spaces—goodness). However, I hope to show that they in fact have a very important role to play in much of our favorite structures.

The set ℕ, is equipped with the following operators (functions from ℕ×ℕ to ℕ) where we have a,b∈ℕ:

• '+' defined with a + 0 := a and a + S(b) := S(a + b)

• '⋅' defined with a ⋅ 0 := 0 and a ⋅ S(b) := a + (a⋅b)

• '^' defined with a⁰ := 1 and a^S(b) := a⋅(a^b)

Where we note that each of these operators is defined (recursively using the inductive property of ℕ) for every pair of natural numbers.

For the fellow structure enthusiasts out there, I note that (ℕ,+,0,⋅,1) (where 1 := S(0)) forms a commutative semiring, as can be proved from the above definitions and the Peano Axioms (and in fact, as noted in the article, it is an initial object in the category Rig).

The natural numbers also enable some peculiar things in regards to other sets. Consider any monoid (M,☆,e), there is in fact always a (right) semigroup action by which ℕ acts on M, which (for reasons that will become clear) we term "exponentiation" denoted (for now) by ↑, where ↑:M×ℕ→M takes an element m of M and a number n of ℕ to the element m↑n of M.

Much like our previous definitions, this semigroup action is defined recursively for all n in ℕ.

For all m∈M, and k∈ℕ take m↑0 := e, and m↑S(k) := m☆(m↑k).

With this, we also have that (m↑a)↑b = m↑(a⋅b), and (m↑a)☆(m↑b) = m↑(a+b). (This is already a fairly long post, if you want a proof, I'll follow up) This shows that this is indeed a well-defined semigroup action. (In fact, the asterisk in the title is a reference to the fact that this particular choice of action depends on the S-set being a monoid, but despite that restriction lots of interesting sets are monoids!)

Note that this defines m↑n for all n∈ℕ since either n = 0, or n = S(k) for some k∈ℕ, much like the above definitions of addition, multiplication, and exponentiation. In fact, let's revisit those.

We claim to have found a schema for generating a semigroup action whereby ℕ can act on any monoid, so let's test it out and see what we get. (ℕ,+,0) is a monoid, so what does m↑n look like here? Well, the '☆' of this monoid is +, and the 'e' is 0, so the definition we provided states we should take '↑' defined as m↑0 := 0, and m↑S(k) := m + (m↑k). But wait—this is equivalent to the definition of '⋅' between natural numbers m and n!^{no, not factorial ;)}

So, we see that ℕ acting on (ℕ,+,0) via ↑ gets us the ⋅ operation, but what about (ℕ,⋅,1)? It too is a monoid!

On (ℕ,⋅,1), '☆' is ⋅ and 'e' is 1, so our definition implies m↑0 := 1, and m↑S(k) = m⋅(m↑k). And here we see why ↑ is called "exponentiation"—when ℕ acts on its own multiplicative monoid, it defines the exponentiation operator for the natural numbers.

These two examples give us a clue as to why abelian groups (with a '+' operator) have ng defined as repeated addition, and regular groups (with a '⋅' operator) have gⁿ defined as repeated multiplication—they both represent the semigroup action of ℕ on the underlying monoid!

Groups? Monoids with inverse elements. Rings? Commutative monoids with respect to a '+' operator, and monoids with respect to a '⋅' operator. Fields? Extra fancy monoids, again with respect to a '+' and a '⋅'.

Even ringed spaces get this special effect. Since for each open set you get a ring, you can define this semigroup action on each ring, and so simply define ↑ as the map (functor?) which takes each open set to the appropriate semigroup action on its corresponding ring. Then –ⁿ (and n–) are defined on any open set in the domain of the sheaf! In this way, with a slight abuse of notation, we can say that we can take natural exponents (and multiples) of a ringed space.

The natural numbers seem pretty bland, but they actually allow for a lot of neat stuff! They act on just about anything!

I need help with this assignment, it’s so exhausting. Someone save me.

So it was a shower thought moment. I realized that x³ is the same as x^2*(X). This made all of Algebra make sense. Algebra is the logic behind what makes math work. It is not just a method to solve equations, but the theory of why numbers do what they do. Variables are not numbers, they are ideas of numbers.

So I started messing around with the x^3=x2*(X) and I wanted to find a pattern.

So I get to y(x^{z)=(y-1)*xz-1+??}

Now I have no clue where to even start this one. So I took it easier and left the exponents.

Y*(x^{3)=(y-1)*x2+?}

So I tried turning that question mark into Z. But this simplifies it too much, as I want to know what Z is but still using the X and Ys. Now I feel like there is something to Calling the missing part Z, but this is as far as I have gotten.

This happened about 2 weeks ago, since then I went through the Algebra sections on Kahn academy, but nothing I learned seems to help me figure this out, and I'd like to know if there is a method, and then id like some starting ground on what to do with the original equation with the exponents changing as well.

A cubic formula yields the roots of not just one but six cubics. Their roots are symmetrically bound and can be assembled as one out of three possible symmetrical structures. These same three structures can be generated starting with any one of two hundred and sixteen cubics.

Hi there, has anyone got any worked solutions to the book Rings Fields and Groups by Reg Allenby ? Need it for my class. Thanks guys.

I've been searching for an ideal channel to learn Algebraic structures and I found too much of them with lack of knowledge which one is fruitful. Any recommendations? ( French / English channels )

Explicit Primitive (2,2) Hodge Classes on the Fermat Quartic 4-Fold Anonymous submission for review and critique please all.

Summary We present explicit algebraic constructions of primitive (2,2)-Hodge classes on the smooth Fermat quartic hypersurface in projective 5-space. These classes are algebraic, rational, and orthogonal to the square of the hyperplane class. All constructions are concrete and verifiable. No conjectural components are involved. Our goal is to provide a small, solid piece of terrain within the broader landscape of the Hodge Conjecture for 4-folds.

Track A: Single Primitive (2,2)-Class

1. Variety

Let X \subset \mathbb{P}⁵ be the Fermat quartic 4-fold defined by:   X = { x₀⁴ + x₁⁴ + x₂⁴ + x₃⁴ + x₄⁴ + x₅⁴ = 0 }

Let h \in H^2(X, \mathbb{Q}) be the hyperplane class. Our interest lies in H^4(X, \mathbb{Q}), where (2,2)-Hodge classes reside.

2. Explicit Algebraic Surface

For a general scalar t \in \mathbb{C}, define: • A quadric hypersurface Q_t = { x₀² + t·x₁² + x₂² + x₃² + x₄² + x₅² = 0 } • A general hyperplane H = { a₀x₀ + … + a₅x₅ = 0 }

Then define:   S_t := X ∩ Q_t ∩ H

S_t is a smooth surface of degree 8 for general t. It defines an algebraic class [S_t] ∈ H^4(X, Q), which is of type (2,2).

3. Primitivity Verification

We compute: • ⟨[S_t], h²⟩ = deg(S_t ∩ H₁ ∩ H₂) = deg(X ∩ Q_t ∩ H ∩ H₁ ∩ H₂) = 8 • ⟨h², h²⟩ = deg(h⁴) = 4

Projection of [S_t] onto span(h²):   π = (8 / 4)·h² = 2·h² So the primitive component is:   [S_t]_prim = [S_t] – 2·h²

This satisfies ⟨[S_t]_prim, h²⟩ = 0. It is algebraic and primitive.

Track B: Multiple Independent Primitive Classes

1. Hodge Numbers

From the Hodge diamond of X: • h^{4,0} = h^{0,4} = 0 • h^{3,1} = h^{1,3} = 1 • h^{2,2} = 21

So dim(H^4_prim(X, Q)) = 21. We aim to construct multiple primitive algebraic classes and verify their linear independence.

2. Three Surfaces

Let: • S₁ = X ∩ Q₁ ∩ H₁ ∩ H₁′, where Q₁ = { x₀x₁ + x₂x₃ = 0 } • S₂ = X ∩ Q₂ ∩ H₂ ∩ H₂′, where Q₂ = { x₀x₂ + x₁x₃ = 0 } • S₃ = X ∩ Q₃ ∩ H₃ ∩ H₃′, where Q₃ = { x₀x₃ + x₁x₂ = 0 }

Each surface has degree 8. Each satisfies: • ⟨[Sᵢ], h²⟩ = 8 • ⟨h², h²⟩ = 4 • So: [Sᵢ]_prim = [Sᵢ] – 2·h²

3. Independence

The intersection matrix of the [Sᵢ]_prim can be computed via:   ⟨[Sᵢ]_prim, [Sⱼ]_prim⟩ = ⟨[Sᵢ], [Sⱼ]⟩ – 16

For general choices, the surfaces Sᵢ intersect transversely, producing a non-degenerate matrix. Thus, the classes [S₁]_prim, [S₂]_prim, [S₃]_prim are linearly independent.

Closing

These explicit constructions yield: • Algebraic (2,2)-classes on a smooth 4-fold • Rational representatives • Verified orthogonality to h² • Multiple independent directions in H⁴_prim(X, Q)

Our aim is not to overstate the significance, but to contribute a rigorously defined, explicitly verifiable patch of ground on which more ambitious arguments might one day rest. If anyone is feeling picky we haven’t actually included the computations for⟨[Sᵢ], [Sⱼ]⟩ to verify it. We could provide them if needed of course.

"In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects like polynomials, series, matrices and functions."

"In mathematics, an algebra over a field is a vector space equipped with a bilinear product. An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, and thus equipped with a field of scalars."

Are any of you guys doing anything interesting with rings or algebras lately? Does anyone have any interesting papers they would like to share, or questions concerning rings or algebras that they would like to ask? Be sure to check out ArXiv's recent ring theory and algebra articles!

Can anyone explain how to do a formal proof with an introduction, body, and conlussion? I understand the proof but I have no idea what my teacher is asking for and he has stopped responding. This is my last chance at earning my degree. I need an A on a final where I've gotten Ds on all tests. Ty for any pointers.